Chiral Limit of 2d QCD Revisited with

Lightcone Conformal Truncation

Nikhil Ananda, A. Liam Fitzpatrickb, Emanuel Katzb, and Yuan Xinc

aDepartment of Physics, McGill University, Montreal, QC H3A 2T8, CanadabDepartment of Physics, Boston University, Boston, MA 02215, USAcDepartment of Physics, Yale University, New Haven, CT 06520, USA

Abstract: We study the chiral limit of 2d QCD with a single quark flavor at finite

Nc using LCT. By modifying the LCT basis according to the quark mass in a manner

motivated by ’t Hooft’s analysis, we are able to restore convergence for quark masses

much smaller than the QCD strong coupling scale. For such small quark masses, the

IR of the theory is expected to be well described by the Sine-Gordon model. We verify

that LCT numerics are able to capture in detail the spectrum and correlation functions

of the Sine-Gordon model. This opens up the possibility for studying deformations of

various integrable CFTs using LCT by considering the chiral limit of QCD like theories.

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Contents

1 Introduction 1

2 2d Chiral Lagrangian and Sine-Gordon 3

3 2d QCD in Lightcone Hamiltonian Truncation 5

3.1 Lightcone Hamiltonian 5

3.2 Massless Quarks Review 5

4 The Challenge at Small Quark Mass 8

4.1 Small Quark Mass and Boundary Conditions 8

4.2 Infinite Nc Warm-Up 10

5 Finite Nc 13

5.1 Boundary Condition Implementation 13

5.2 Determining the Correct Boundary Condition 15

6 Comparison of Truncation Results with Chiral Lagrangian 17

7 Conclusions and Future Directions 22

A Heff Derivation of Induced Gauge Interaction 24

B Details of Modified Basis 25

C Form factors in Sine-Gordon theory 30

1 Introduction

Understanding gauge theories in the non-perturbative regime is a worthy goal for many

applications to both condensed matter and in particle physics. For the most part lattice

gauge theory (LGT) has been the preferred tool to study gauge theories. Nevertheless,

LGT is not ideal when it comes to capturing real time dynamics, and faces complications

in maintaining certain symmetries, such as chirality or supersymmetry. Hamiltonian

truncation methods, on the other hand, allow for direct access to time evolution, and

– 1 –

as they do not discretize space, can often preserve some subset of the symmetries. In

the 2d context, there has been much work studying gauge theories using lightcone (LC)

quantization, where one can choose lightcone gauge and still maintain gauge invariance

in the presence of a hard LC momentum cutoff. In this context, most of the work has

been done using Discrete Lightcone Quantization (DLCQ)[1].

A model which remained out of reach, however, was a 2d version of real 4d QCD,

where the strongly bound quarks have nonvanishing masses much lighter than the

strong coupling scale (set in 2d by the gauge coupling g). As we will review below,

though it is possible to accommodate either strictly massless quarks or heavy quarks [1–

8], for light quarks, bound state wavefunctions develop features which can be difficult to

approximate. Consequently, DLCQ as well a generic Lightcone Conformal Truncation

(LCT) approach will lead to slow convergence. Studying QCD in the light mass, or

chiral limit, is desirable not only because of its closer resemblance to its 4d cousin, but

also because the effective theory describing the low energy degrees of freedom of QCD

like theories can be very rich in its own right. Indeed, recently, Delmastro, Gomis

and Yu [9] have offered a full classification of the IR of QCD theories in the chiral

limit, building on the earlier insights of [5, 10, 11]; examples include minimal models

and WZW models. Hence, a method which can efficiently capture the chiral limit of

QCD like theories, can also be used to study relevant deformations of a wide class of

integrable CFTs.

In this paper, we take the first step towards exploring the chiral limit by consid-

ering the simplest case of 2d QCD with a single quark using LCT. Anomaly matching

requirements constrain the IR of this theory to be that of a single real scalar “pion”.

By standard arguments, a mass deformation results in a cosine potential for the pion.

This is of course the well-known Sine-Gordon model, for which many results are known

from integrability. This setting is thus ideal to check LCT and its ability to capture

the c = 1 degree of freedom.

We begin in section 2 with a review of the connection between the chiral Lagrangian

of QCD and the Sine-Gordon model in a somewhat more modern language. In section 3,

after reviewing the lightcone formulation of QCD, we first focus on the strict massless

limit, explaining why the naive LCT basis contains an exactly massless sector with

c = 1 at finite Nc. Next, in section 4 we review the challenge posed by light quarks,

using the large Nc limit as warmup. In this limit, we describe a change to the LCT basis

in terms of the quark mass (as motivated by ’t Hooft’s analysis [12]) which restores fast

convergence. In section 5 we generalize the modified LCT basis to multi-particle states

(essential to solve the theory at finite Nc). We find that the necessary modification

of the LCT basis agrees well with the earlier two-particle approximation suggested

by Sugihara et al. [13], especially for very small quark masses. Finally, in section 6

– 2 –

we report on our numerical results for the spectrum and for correlation functions for

Nc = 3, and compare them to detailed expectations from the Sine-Gordon model. We

find very good agreement between numerics and integrability in the limit of very small

quark masses. In particular, we reproduce numerically the c-function (or stress-tensor

correlation function) of the Sine-Gordon model in Fig. 10.

2 2d Chiral Lagrangian and Sine-Gordon

Before we consider 2d QCD in Hamiltonian truncation, we will first determine what

the spectrum should look like in the limit of small quark mass mq by studying the

chiral Lagrangian for the pion. It turns out that in this limit, the low-energy theory is

described by the Sine-Gordon model [14]. Here, we will rederive this result in modern

terms. We begin with the statement that the axial symmetry of the theory is nonlinearly

realized by shifts on the pion. That is,

qq ∼ eiπ/fπ , (2.1)

so the axial symmetry q → eiαγ5q acts on π by π → π + 2αfπ. Therefore, at mq = 0,

the chiral Lagrangian is simply L = 12(∂π)2, plus irrelevant interactions. When mq is

nonzero but small, we can treat it as a spurion for the axial symmetry and add the

following invariant combination to the action:

L ⊃ b(mqΛe

iπ/fπ +m∗qΛe−iπ/fπ

), (2.2)

where Λ is the confinement scale, and b is an unknown dimensionless constant.

Remarkably, in 2d we can fix fπ in terms of Nc using anomaly matching. In terms

of the pion field, the axial current is

JµA = 2fπ∂µπ, (2.3)

so that [i∫dxJ0

A(x), π(y)] = 2fπ. To relate fπ and Nc, we compute the axial anomaly

coefficient in the IR and in the UV. In the IR at zero mq,

〈JµA(p)JνA(−p)〉 = 4f 2π

pµpνp2

, (2.4)

whereas in the UV the quark loop contributes

〈JµA(p)JνA(−p)〉 ⊃ Nc

π

pµpνp2

. (2.5)

– 3 –

Matching the UV and IR anomaly coefficient, we obtain the relation

f 2π =

Nc

4π. (2.6)

Putting everything together and taking mq to be real, we identify the chiral Lagrangian

as the sine-Gordon model,

L =1

2(∂π)2 + λ cos βπ, (2.7)

with

λ = bmqΛ, β = 2

√π

Nc

. (2.8)

It is well-known that due to quantum effects, the scaling dimension of the cosine oper-

ator is

∆[cos βπ] =β2

4π=

1

Nc

, (2.9)

and so is relevant for physical values of Nc. There are also subleading interactions with

additional powers of mq, the next one being ∼ m2q cos(2π/fπ). By equation (2.9), the

dimension of this subleading interaction is ∆ = 4Nc

.

The sine-Gordon model in 2d is integrable, and the mass spectrum of states is

known in closed form [15]. Our main interest will be in a handful of the lightest states.

These are a baryon, a meson, and a bound state of two mesons, with masses mB,mM

and m2 respectively. They are related to each other by

mM = 2mB sin

(π

2(2Nc − 1)

), m2 = 2mB sin

(π

2Nc − 1

). (2.10)

Expanding in 1/Nc,

m2

2mM

≈ 1− π2

32N2c

+O(N−3c ), (2.11)

so the bound state mass is very close to 2mM even for modestly large Nc; in particular,

at Nc = 3 [Nc = 2], m2

2mM= 0.95 [0.87].

– 4 –

3 2d QCD in Lightcone Hamiltonian Truncation

3.1 Lightcone Hamiltonian

Our main focus in this work is to apply LCT to two-dimensional QCD with fundamental

matter, given by the Lagrangian

L = −1

2TrF µνFµν + Ψ(i /D −mq)Ψ. (3.1)

Our goal is to find the eigenvalues and eigenvectors of the invariant mass-squared

operator M2 = 2P+P−, working in lightcone quantization, i.e. quantizing on surfaces

of constant lightcone “time” x+ where

x± ≡ x0 ± x1

√2

. (3.2)

A particularly advantageous gauge choice is lightcone gauge, A− = 0. The Hamiltonian

P+ in this gauge is straightforward to derive by integrating out the nondynamical

degrees of freedom. Briefly, one separates the fermion field Ψ into left- and right-

movers

Ψi =1

21/4

(ψiχi

)(3.3)

where i is the fundamental index which we will suppress from now on to avoid clutter.

The Lagrangian then takes the form

L = i(ψ†∂+ψ + χ†∂−χ) + gψ†AA+TAψ + Tr(∂−A+)2 − mq√

2(χ†ψ + ψ†χ). (3.4)

The right mover χ and the gauge field A+ are non-dynamical, and can be integrated

out to obtain the following form of the Hamiltonian:

P+ =

∫dx−T−+ =

∫dx

[m2q

2:ψ†

1

i∂ψ:− g2

2:ψ†TAψ:

1

∂2:ψ†TAψ:

], (3.5)

where we have suppressed − subscripts on the derivatives (which will be implicit from

now on) and the superscript on the spatial coordinate x−.

3.2 Massless Quarks Review

The presence of a small but nonzero quark mass mq � g presents challenges that will

be the main focus of this work. To warm up, we first review the massless case, and

describe the advantages of the LCT approach here.

– 5 –

At mq = 0, the only interaction is the quartic term in (3.5). The gauge boson

propagator 1∂2

is ambiguous and must be defined more precisely; ’t Hooft [12] defined

it in momentum space as the following ‘principal value prescription’:

1

∂2→ P.V.

1

k2−≡ 1

2

(1

(k− + iε)2+

1

(k− − iε)2

). (3.6)

This definition is manifestly finite (eg by contour deformation) when integrated against

any smooth function. In appendix A, we show that ’t Hooft’s principal value condition

for the nonlocal interaction can be derived by using the lightcone effective Hamiltonian

Heff from [16].

The interaction as written in (3.5) is not normal-ordered in the usual sense, but for

convenience it can be separated into a normal-ordered operator and the contribution

from the internal contractions of the fermions. These internal contractions are easily

evaluated and produce a finite shift in the fermion mass term operator, ∼ ψ† 1∂ψ.

:ψ†TAψ:1

∂2:ψ†TAψ: ∼= :ψ†TAψ

1

∂2ψ†TAψ:− C2(Nc)

π:ψ†

1

∂ψ: (3.7)

where C2(Nc) = N2c−1

2Nc. An important effect of the second term in (3.7) is that the the

state created by the axial current remains exactly massless when the bare quark mass

mq vanishes, due to a cancellation between the two terms in (3.7). One can also reverse

this logic, and use the fact that anomaly-matching requires the current to create a

massless state in order to fix the coefficient of the second term.

Although representing the interaction as the sum of normal-ordered terms in (3.7)

is convenient for the Fock space representation of the states, it obscures the conformal

structure of the interaction and in particular the fact that the dynamics of the interact-

ing theory are controlled by the current algebra of the theory and its representations.

This fact and some of its implications were explained in [11] and have been exploited

many times since (eg [17]). It is especially advantageous in LCT, where the main point

of the LCT basis is that all basis states are simply Fourier transforms of local CFT

primary operators acting on the vacuum:

|Oi, p〉 ≡1

NOi

∫ddxe−ip·xOi(x) |vac〉 , (3.8)

with NOi a normalization factor. This connection between the states and primary

operators allows one to take advantage of the conformal structure of the UV fixed

– 6 –

point. More explicitly, the basis states take the form [8]

|O, p〉 =1

NO

∫dx e−ipxO(x)

=1

NO

∫dx e−ipx

∑k

Ck

(∂k11ψ†i1∂

k21ψi1

)(∂k12ψ†i2∂

k22ψi2

)· · ·(∂k1nψ†in∂

k2nψin

)(x) |vac〉 ,

(3.9)

where NO is a normalization factor and where in the second line, we have expanded the

operator O(x) in terms of its “monomial” constituents. The Ck coefficients are chosen

such that the states (3.9) form a complete, orthogonal basis.1

To see how the spectrum is controlled by the Kac-Moody (KM) current algebra,

recall first that in the UV, the Nc fermions have a U(Nc)1 current algebra, and its

SU(Nc)1 subgroup is gauged. We can write the U(1) current in U(Nc)1 as J0 ∼ ψ†iψi,

and the SU(Nc)1 currents as Ja ∼ ψ†iTaijψj. All operators with vanishing baryon number

can be constructed out of products of Ja, J0 and their derivatives.2 In this context, it is

more convenient to define composite operators by taking contour integrals, ie by using

the product (AB)(z) ≡∮

dw2πi

A(z)B(w)z−w . The reason this is more convenient is that now

the matrix elements of the Hamiltonian are all manifestly just integrals of correlators

of currents, and correlators of currents are completely fixed by the algebra.

The massless sector is particularly simple in this description: the massless states

are just those made from products of J0 and its derivatives. To see why, note that

the interaction Ja 1∂2Ja is simply a (double) integral over two insertions of the SU(Nc)1

currents, so we can extract the Hamiltonian matrix elements from correlators with an

insertion of Ja(y)Ja(y′). By holomorphicity, correlators of currents are fully determined

by their poles, so a correlator of the form 〈O(x)Ja(y)Ja(y′)O′(z)〉 vanishes if it has no

poles as a function of y′.3 Therefore, states made from primary operators O without

singular terms in the Ja(y)O(0) OPE are massless. In particular, correlators of the

form 〈J0(z1) . . . J0(zn)Ja(y)Ja(y′)〉 vanish. Since the matrix elements of states built

1In Fock space language, the conformal operator basis used in LCT simply corresponds to takingmulti-quark states with wavefunctions that are polynomials in momentum for the individual quarkcomponents. For example, a general two-quark singlet state can be written as

|Ψ, p−〉2 ≡∫ p−

0

dq−f(q−)|q−, a; p− − q−, a〉. (3.10)

summation over a = 1, . . . , N implied. The massless pion is just a two-quark singlet with a constantwavefunction, f(q−) = const.

2For instance, the stress tensor is T = 12 (∂ψ†iψi − ψ†i ∂ψi) = 1

2Nc(J0J0)(z) + 1

2(Nc+1) (JaJa)(z).

3The contraction of Ja with itself, Ja(y)Ja(y′) ∼ N2c−1

(y−y′)2 , is a bubble diagram that is removed in

lightcone quanitzation.

– 7 –

from powers of J0 are just integrals over such correlators, they also vanish, and the

massless states in LCT are simply the primaries that can be constructed out of J0(z),

without using Ja.

Returning to the massive sector of the theory, we can ask how to identify the

massive single particle states. This is not as simple as it might sound since massive

bound states appear in the midst of a multi-particle continuum due to the pions, and

therefore do not particularly stand out if one looks at the spectrum alone. A useful

way to identify the bound states is to look at the spectral density of the stress tensor.

Due to anomaly matching, massless particles in 2d cannot have interactions and only

contribute to T−− through a kinetic term in the IR that looks like ∼ (∂π)2. Thus, T−−only has overlap with massive bound states, and the lightest massive single-particle

mesons appear as isolated poles in the spectral density. At higher energies, T−− also

has overlap with the multi-particle continuum of these massive particles.

Concretely, the spectral density of T−− can be obtained by summing over overlaps

ρT−−(µ) =∑j

|〈T−−(0)|µj, p〉|2δ(µ2 − µ2j). (3.11)

For example, Fig. 1 shows the spectral density of the T−− component of the stress-

energy tensor at ∆max = 15 which corresponds to 3,032 basis states for Nc = 3. This

is shown in Fig. 1, where we can read off the first few massive meson states.

4 The Challenge at Small Quark Mass

4.1 Small Quark Mass and Boundary Conditions

In this section we will discuss the regime where the fermion mass term

δLmass = −m2qψ† 1

∂ψ =

∫dp

8π(a†pap + b†pbp)

m2q

2p(4.1)

is small but nonzero.

The presence of a quark mass term forces one to adjust the basis of states to

accommodate a new boundary condition for the wavefunctions. To see this, consider

the mass term matrix element of an arbitrary 2-particle state

〈∂k1ψ†∂k2ψ, p|M |∂k′1ψ†∂k′2ψ, p′〉 = (2π)δ(p− p′)×∫dp1

8πpk1+k′11 p

k2+k′22

(m2q

2p1

+m2q

2p2

)p2=p−p1

.(4.2)

– 8 –

0 50 100 150 20010-40

10-30

10-20

10-10

1

mass eigenvalue μ2

spectraldensity

ρT--(μ)

Figure 1: Shows the spectral density of T−− at a truncation cutoff of ∆max = 15, whichcorresponds to 3,032 basis states at Nc = 3. The massless sector does not contributeto the spectral density due to anomaly matching. Massive states do contribute to thespectral density and thus appear as isolated poles above this sea of states with masslessparticles. We use units where g2Nc/π = 1.

Due to the 1pi

factors, the integral is divergent at the pi → 0 or pi → p boundaries,

if either k1 = k′1 = 0 or k2 = k′2 = 0. The consequence of this IR divergence is that

some states become infinitely massive and are lifted out of the IR spectrum. The states

that remain in the spectrum are those with wavefunctions that vanish at pi → 0 for

any of the individual fermion momenta. The treatment of this type of IR divergence

is discussed in [7]. A complete basis of states can be made out of local operators that

are products of fermions where each fermion has at least one derivative acting on it;

[7] refer to this as the “Dirichlet” basis. Each such state vanishes like ∼ pi as pi → 0.

As long as mq is not too small, this Dirichlet basis is sufficient for practical cal-

culations. However, when mq/g � 1, although this basis is complete, it experiences

extremely slow convergence with ∆max. The reason for the slow convergence is that

the true eigenstates of the theory vanish at pi → 0 like pαi for some small value of

α. Accurately approximating this nonanalytic behavior near the boundary requires a

very large number of basis states that individually behave analytically. In the next

subsection, we review this phenomenon at infinite Nc as a warm-up to our treatment

at finite Nc.

– 9 –

4.2 Infinite Nc Warm-Up

Consider QCD with massive quarks in the large-Nc limit. We normalize the gauge

coupling to one, g2Nc/π = 1, so the theory depends only on the parameter mq. In

this limit, the particle-number-changing interaction is suppressed by 1Nc

, allowing us to

simplify our basis by only including 2-particle states. We can take the truncation ∆max

high enough to quantitatively test the convergence and illustrate all of the subtleties

involved with nonzero mass. We will see that the LCT setup with the Dirichlet basis

has good convergence if the quark mass is not too small (mq & 0.2g√Nc/π). For small

fermion mass, however, the boundary condition imposed by the Dirichlet basis leads to

slow convergence, because it is trying to approximate pα with a polynomial in p.

The boundary condition of the large Nc QCD momentum space wave function was

solved analytically in [12]. In the limit that either parton’s momentum approaches

zero, the wave function is a power law pα∗i , where the power α∗ is determined by the

quark mass through

πα∗ cot (πα∗) +m2q = 1 . (4.3)

The Ansatz is then chosen to satisfy this boundary condition. The numerical values of

α∗ as a function of mq is plotted in Figure 2. The true boundary condition smoothly

��� ��� ��� ��� ��� ��� ��� ���

���

���

���

���

���

���

��

α*

Figure 2: The numerical values of α∗ in the boundary asymptotic behaviorf(pi, · · · ) ∼ pα∗i , for the large Nc QCD states, according to (4.3).

interpolates between zero and O(1) values. In comparison, the LCT non-Dirichlet basis

has α = 0 and the Dirichlet basis has α = 1.

In principle, one can choose any value of α with 0 ≤ α ≤ 1 and then define a set

of LCT basis states that vanish like pαi at pi ∼ 0. While all values of α > 0 will satisfy

the Dirichlet boundary condition f(0, · · · ) = 0 and thus work in principle, we expect

– 10 –

only the ones close to ‘t Hooft’s solution (4.3) will give the variational Ansatz a fast

convergence rate.

As a particularly illustrative example of this slow convergence rate for the “wrong”

choice of α, consider the case where we use the Dirichlet basis (i.e. α = 1). The Dirichlet

basis states and their matrix elements are described in detail in [7]. At each value of

mq (in units where g2Nc/π ≡ 1), we diagonalize the Hamiltonian and obtain the mass

eigenvalues. Since LCT is a variational ansatz, we expect the low-lying spectrum to

converge to the physical values at sufficiently high `max. We then plot these low-lying

eigenvalues as a function and compare them against the known data computed from ‘t

Hooft’s ansatz. The result is shown in 3.

● ● ●●

●●

●●

●●

●●

●●

●●

●●

●●

●● ● ● ● ● ● ● ●●

●●

●●

●●

●●

●●

●●

● ● ● ● ● ● ● ● ●

▲

▲▲

▲▲

▲▲

▲▲

▲▲

▲▲

▲▲

▲▲

▲▲

▲▲

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

Quark mass mq

eigenvaluesmi

ℓmax = 512

Figure 3: Low meson mass eigenvalues mi of the massive quark QCD in the large-Nc

limit, as a function of the quark mass mq. The points are the LCT results computedwith the Dirichlet basis at `max = 512. We use colors red, blue and green for the lowest,second lowest and third lowest eigenvalues, respectively. The black triangles are theknown results using large Nc wave functions with ‘t Hooft’s boundary condition.

The result shows that the LCT result agrees well with the known data (black

triangles) formq & 0.2. However, in the smallmq regime the LCT result starts deviating

from the known result, and its lowest eigenvalue is nonzero at mq = 0 even for the large

truncation value `max = 512. We expect that the error in the lowest eigenvalue should

approach zero with increasing `max at a rate ∼ `−α∗max , so that as mq decreases, the

convergence rate becomes worse and worse and eventually at mq = 0, where α∗ = 0,

the result using the Dirichlet basis never converges to the right answer. In fact, we

see similar slow convergence in DLCQ with the truncation parameter K. In Fig. 4, we

explicitly show the convergence rate of the lowest eigenvalue to its asymptotic value as

– 11 –

a function of K. For a range of quark masses, the convergence rate can be seen to be

K−α∗ , as expected based on the behavior of the true lightest state wavefunction.

0.0 0.1 0.2 0.3 0.4

0.6

0.7

0.8

0.9

1.0

1.1

K-r

mgap

mq 0.25r 0.223554mgap

(LCT) 1.131

●

●

●

●

0.005 0.010 0.050 0.100

0.005

0.010

0.050

0.100

boundary condition α

DLCQconvergencepowerr

r α

● DLCQ data

Figure 4: Measurements of the convergence DLCQ at large Nc at small mq. Left:An example of DLCQ convergence at mq = 0.25. The y-axis is the mass gap. The x-

axis is the truncation K to power −r. We use a fit m(DLCQ)gap (K) = cK−r+mgap(∞) and

take mgap(∞) ≈ m(LCT)gap (∞) from the LCT extrapolation described by Figure 6. Right:

Compare r in the the DLCQ convergence power law K−r with the boundary conditionα computed from (4.3). The DLCQ data comes from repeating the procedures in theleft plot to various mq and obtain the power law fit. A simple function r = α (blackdashed line) fits the relation between r and α quite well.

Our strategy will be to introduce nonlocal operators designed to behave like pαi at

small pi. In the new basis that we call modified boundary condition basis, the building

blocks are the generalized free fields ∂αψ. Their correlators are defined in terms of the

following two-point function:

〈∂αψ(x)∂αψ(y)〉 =Γ(2α + 1)

(x− y)2α+1. (4.4)

Higher n-point functions are defined in terms of the two-point function through Wick

contractions. We relegate the details of how to efficiently compute all the required

Fourier transforms of correlators to Appendix B.

We test the basis by computing the large Nc QCD spectrum at different quark mass

mq, and for each mq we compute the truncated Hamiltonian using modified boundary

condition basis at several different α values around the theoretical value α∗. We then

compare these different results with the theoretical values of from ‘t Hooft’s boundary

condition. The result is shown in Figure 5. The result verifies that the convergence

improves as the boundary condition α of the basis approaches α∗, and that the modi-

fied boundary condition LCT result matches the theoretical result. We also show the

– 12 –

convergence of the lowest mass eigenvalue as a function of the truncation order `max,

taking different α∗, for some small mq. This is shown in Figure 6. When using the

correct boundary condition, the LCT method has a convergence rate of `−5.5max , which is

significantly better than the Dirichlet basis.

10-4 10-3 10-2 10-110-2

10-1

100

boundary condition α

lowestmasseigenvalueM12

m 0.0039

m 0.0156m 0.0625

known result

Figure 5: Plot of the lowest eigenvalue, M21 as a function of the boundary condition

of the LCT generalized free field basis, α. The red, blue, green lines represent resultscomputed using three different quark masses mq, respectively. The black dashed lineis the known result given by the parametric line (α(mq),M

21 (mq)) where α(mq) is ‘t

Hooft’s boundary condition (4.3) and M21 (mq) is the correct lowest eigenvalue. The

known result line intersects each LCT line at the correct boundary condition withrespect to its quark mass.

5 Finite Nc

In the last section we saw that the LCT method successfully reproduces the large Nc

known results. In this section we will use the LCT method to study finite Nc QCD.

With Nc finite, particle number-changing matrix elements are no longer suppressed by

large Nc and we must include multiparticle states.

5.1 Boundary Condition Implementation

As is discussed in the previous section, the fermion wave function has a power law

boundary condition as the individual parton momentum fraction pi/ptot → 0, and at

small quark mass the convergence of truncation results is good only if we choose an

ansatz with the correct boundary condition. At finite Nc, the number of states grows

rapidly with the increasing particle number, so we need to compute the basis and matrix

elements efficiently.

– 13 –

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲

●●

●●

●● ● ●

▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼

6 8 10 12 14 16 18 20

10-10

10-7

10-4

0.1

ℓmax

mgap

2(ℓmax)-mgap

2(∞

)

~ ℓmax-5.49

▲ α 0.500

▲ α 0.137

▲ α 0.0344

● α 0.00861

▼ α 0.00215

▼ α 0.000538

▼ α 0.000135

▼ α 0.0000337

Figure 6: Compare the convergence of the mass gap using different boundary condi-tion α. We take mq = 0.015625 and vary the boundary condition α. For each α, wediagonalize the Hamiltonian and obtain the mass gap mgap and fit it as a function of`max. The fit model is a power law m2

gap(`max) = m2gap(∞) + c1`

−c2max, with parameters

m2gap(∞), c1 and c2 to be determined by the fit. The boundary conditions α have a

drastic effect on the convergence rate. The optimal boundary condition α∗ = 0.00861(blue line), which agrees with (4.3). There the convergence is very fast, ∼ `−5.5

max . Awayfrom α∗ (points of other colors), the convergence is poor.

We implement the full-fledged LCT method with a generalized free field basis. In

the process of building the basis, we treat ∂αψ as a primary operator, where α is a

real number which we will determine later using adaptive methods. The full basis is

constructed recursively by sewing two lower dimension primary operators A and B to

form a double-trace operator

[AB]` ≡∑m=0

c`m(∆A,∆B) ∂mA∂`−mB, (5.1)

where the coefficients c`m(∆A,∆B) are given by the formula

c`m(∆A,∆B) =(−1)mΓ(2∆A + `)Γ(2∆B + `)

m!(`−m)!Γ(2∆A +m)Γ(2∆B + `−m). (5.2)

We find selection rules to optimize the efficiency of constructing the basis in the charge-

neutral sector. We begin with bosonic operators φn ≡[∂αψ†∂αψ

]n. Each φn spans a

basis of multi-boson primaries similarly to that of a free scalar, and the full basis

is spanned by taking arbitrary multi-trace operators between primaries in different φn

– 14 –

bases. At large Nc this basis is the full minimal basis since all φn are truly independent.

Finite Nc reduces the space a bit more, and we need to compute the Gram matrix to

eliminate the redundancy due to algebraic constraints. Because primaries at different

dimensions are automatically orthogonal, we only need to compute the Gram matrix

within each dimension sectors. When we compute the matrix elements, we decompose

each primary operator into a linear combination of “monomials”

O ≡∑k

COk (∂k1+αψ†∂k2+αψ) · · · (∂k2n−1+αψ†∂k2n+αψ), (5.3)

and we need to compute the matrix elements between every pair of monomials. The

space of monomials is more redundant than that of the primary operators, because

they also span all descendant states. In momentum space, the descendant states are

not linearly independent from the primary states, which means we can use the Gram

matrix to eliminate the redundant monomials. Finally, we use the Wick contraction to

compute the matrix elements. The Wick contraction is much more efficient than the

Fock space integrals as it reduces the integrals between every pair of partons to one

expensive principal value integral of only one pair of variables dx and dx′ and leaves the

spectator partons as a simple algebraic expression that can be calculated recursively.

5.2 Determining the Correct Boundary Condition

We will use a variation method to determine the correct boundary condition α numeri-

cally. Hamiltonian Truncation is a variational ansatz, hence the ground state eigenvalue

is always overestimated by finite truncation. We will take α as the variational param-

eter, and change α adaptively until we find the value α∗ that minimizes the mass gap.

An approximation α∗ ≈ α0 =√

3mq√1+N−2

c π+ O(m2

q) can be found [13] by restricting to

two-particles, and solving the Bethe-Salpeter equation with finite Nc at small parton-

x.4 We will take α0 as the initial seed of the search. The result is shown in Figure 7.

The good news is that α∗ converges fast enough so that we can perform the expensive

variational search at low ∆max and use the same α∗ to compute the matrix elements

at high ∆max. Once we have chosen an optimal (or nearly optimal) value of α for the

construction of our basis, the gap converges reasonably (∼ ∆−1max) as a function of ∆max,

as we show in Fig. 8.

4More generally, the full expression for α0 is given by the solution to the equation πα0 cot(πα0) +m2

q

1−N−2c

= 1, see [13]. We emphasize that α0 is defined as the boundary condition in the two-particle

truncation, whereas α∗ is defined as the true boundary condition in the full theory.

– 15 –

0.001 0.005 0.010 0.050 0.100 0.500

0.95

0.96

0.97

0.98

0.99

1.00

mq

α*/α0

Δmax 4

Δmax 6

Δmax 8●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

0.01 0.1 1

1.×10-4

5.×10-40.001

0.0050.010

0.050

1-α*/α0

Figure 7: A plot of the optimal boundary condition α∗ determined by the variatioalmethod at different truncation ∆max, normalized by the initial approximation α0. Atsmall mq, α∗ has already perfectly converged at small ∆max.

0 1

4

1

6

1

8

1

10

1

12

0.003115

0.003120

0.003125

0.003130

0.003135

0.003140

0.003145

0.003150

Δmax-1

mgap

2

mq 0.001

0 1

4

1

6

1

8

1

10

1

12

0.0620

0.0625

0.0630

0.0635

0.0640

0.0645

0.0650

Δmax-1

mgap

2

mq 0.02

0 1

4

1

6

1

8

1

10

1

12

0.335

0.340

0.345

0.350

0.355

Δmax-1

mgap

2

mq 0.1

0 1

4

1

6

1

8

1

10

1

12

2.540

2.545

2.550

2.555

2.560

2.565

2.570

Δmax-1

mgap

2

mq 0.5

Figure 8: A plot of the convergence of the mass gap at different mq. In all four cases,the mass gap has converged to percent level accuracy, and even better for mq = 0.001.

– 16 –

6 Comparison of Truncation Results with Chiral Lagrangian

Now we have addressed all the conceptual and technical issues in studying massive

2D QCD at finite Nc using LCT. In this section we present these numerical results

and, as promised, compare the truncation data at small mq with the Chiral Lagrangian

prediction. In section 2 we argued that the Chiral Lagrangian describes the Sine-Gordon

model (2.7), which means we know the spectrum exactly at small mq.

At Nc = 3, we expect 5 bound states, two of which have mass below the two

particle threshold. Figure 9 shows the spectrum and the c-function of three different

choices of mq. For the mq = 0.001 and mq = 0.02 spectrum plots, the three highlighted

eigenvalues (red, blue, green, respectively) converge to the Sine-Gordon prediction val-

ues to a good accuracy, while the mq = 0.1 the same eigenvalues have a noticeable

deviation. This observation is consistent with the c-function plot. Both mq = 0.001

and mq = 0.02 c-functions have a clear plateau at c = 1, which matches the central

charge of the free massless scalar. The mq = 0.02 c-function has a small correction to

the plateau central charge, and the UV states enter earlier than that of mq = 0.001.

The mq = 0.1 c-function has almost no plateau. We conclude that mq = 0.001 is small

enough to match to Sine-Gordon in the IR. By contrast, Sine-Gordon gets a small

correction at mq = 0.02 and a large correction at mq = 0.1.

nmax 4 6 8 no limit Sine Gordonm1 0.0033341 0.00312532 0.0031251 0.00312507 −m2 0.0126691 0.0115498 0.0113329 0.0113315 −

m2/m1 1.94932 1.92238 1.90431 1.90421 1.90211Ebind/m1 0.05068 0.07762 0.09568 0.09579 0.09789

Table 1: Binding energy Ebind ≡ 2m1 − m2 of the second bound state measuredfrom the LCT Hamiltonian restricted to nmax particles. We use truncation ∆max = 12and small quark mass mq = 2−10 ≈ 0.001. The LCT result accurately reproduces theSine-Gordon theory prediction of the binding energy for nmax = 8 or higher, whilethe measurement with particle number restricted to 4 has a significant error. Our∆max = 12 basis has 77 states in the light sector (i.e. become massless as mq → 0) and741 in the heavy sector.

Now we take a closer look at the small mq regime. We take a tiny quark mass,

mq = 2−10 ≈ 0.001. At this mq the IR data is dominated by Sine-Gordon and the

finite mq correction is negligible, so we are able to compare the LCT result and the

Sine-Gordon result quantitatively. In Table 1 we show the ratio between two lowest

bound state masses m2/m1, and compare it with the ratio m2/mM from the Sine-

Gordon prediction (2.10). Our full-fledged truncation result at ∆max = 12 reproduces

– 17 –

0 1

12

1

10

1

8

1

6

1

4

0.000

0.005

0.010

0.015

0.020

0.025

Δmax-1

mi2

mq 0.001

m22 3.62m1

2

4m12

δm22 ∼ Δmax

-4.11

δmthreshold2 ∼ Δmax

-3.07

0 1

12

1

10

1

8

1

6

1

4

0.0

0.1

0.2

0.3

0.4

0.5

Δmax-1

mi2

mq 0.02

m22 3.62m1

2

4m12

δm22 ∼ Δmax

-2.06

δmthreshold2 ∼ Δmax

-2.97

0 1

12

1

10

1

8

1

6

1

4

0.0

0.5

1.0

1.5

2.0

2.5

Δmax-1

mi2

mq 0.1

m22 3.62m1

2

4m12

δm22 ∼ Δmax

-1.36

δmthreshold2 ∼ Δmax

-2.76

20 40 60 80 100 1200.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

μ2/mgap2

c(μ)

mq = 0.001

mq = 0.02

mq = 0.1

Figure 9: Spectrum and c-function of QCD at Nc = 3 for different quark massesmq. Upper left: The mass eigenvalues as a function of the truncation ∆max, atmq = 0.001. The lowest three eigenvalues are highlighted with red, blue and greenpoints, respectively. We fit the blue and green data at even ∆max to m2 = c1 + c2∆−c3max

and show the fit as blue and green lines, respectively. The red line is mgap(∆max = 12),and its convergence is shown separately in Figure 8. These trend lines predict theeigenvalues in the ∆max → ∞ limit. The gray points are higher eigenvalues. Theblack dashed line is the two particle threshold, and the brown dashed line is the Sine-Gordon theoretical prediction of the second bound state. The second eigenvalue (blueline) clearly deviates from the two particle threshold (black dashed line) and convergesto the Sine-Gordon prediction (brown dashed line) to a good approximation. Upperright: The mass eigenvalues as a function of the truncation ∆max, at a larger quarkmass mq = 0.02. Lower left: The mass eigenvalues as a function of the truncation∆max, at a larger quark mass mq = 0.1. Lower right: The c-function (y-axis) ofthe three quark masses shown in the same plot, using the mass gap as the unit of theenergy scale (x-axis). The blue, red and green lines correspond to mq = 0.001, 0.02 and0.1, respectively. At small mq the c-function has a clear plateau at c = 1, and the UVcorrection sets in at lower scale for increased mq.

– 18 –

the Sine-Gordon binding energy to about 2% accuracy.5 Because the binding energy

itself is small – about 5% – this level of accuracy in the binding energy required a

significantly higher level of accuracy, of about 0.1%, in the total energy of the bound

state. The inclusion of high-particle-number states was absolutely necessary for this

result. The prior work [13] used a different truncation similar to LCT but where the

number of particles was limited to 4 or less. As one can see from the above table, with

at most 4 particles, the binding energy has a significant deviation from the Sine-Gordon

prediction, by about a factor of 2.

●

●

● ● ●●

● ●● ●● ● ● ● ●●● ●●●●●● ● ● ●● ● ● ● ●●

0 10 20 30 40

0.5

0.6

0.7

0.8

0.9

1.0

1.1

μ2/mgap2

c(μ)

● LCT Raw Data

Pade (4,4)

SG c2+c4+cMM

4 6 8 10 12-0.015-0.010-0.0050.0000.0050.0100.015

c Pade-c SG

Figure 10: The IR c-function at mq = 2−10 ≈ 0.001. The discrete points connectedby the dashed line is the LCT raw data computed using (3.11). The red solid line isthe interpolation using analyticity and (4, 4)-Pade approximant. The blue dashed linerepresents the Sine-Gordon integrability result including the contribution from the piontwo particle states cMM , and the one particle states of parity even breathers, c2 and c4.The cMM contribution is continuous, and c2 and c4 are the two kinks at m2

2 = 3.61m2gap

and m24 = 9.47m2

gap; the small spike-like feature in the inset is due to a sub-percent leveldifference in m2

4 between the truncation and Sine-Gordon calculation. The Sine-Gordonresult is accurate up to the soliton-anti-soliton threshold µ2 = 10.47m2

gap. Up to thisenergy, the LCT result follows the Sine-Gordon to a very good accuracy. Beyond thethreshold, one should also add soliton-anti-soliton contributions to the Sine-Gordonc-function, which are not included in the plot above.

For any mq, we can compute the spectral density of the stress tensor, and use

it to compute the Zamolodchikov c-function. At small mq, we can compare the c-

function quantitatively between the Sine-Gordon prediction and the truncation results.

On the Sine-Gordon side, the c-function can be constructed from the form factors

5At ∆max = 4, 6, 8, 10 and 12, the number of (light,heavy) states is (5, 7), (11, 29), (22, 95), (42, 277),and (77, 741), respectively.

– 19 –

of the stress tensor. The larges contribution comes from the meson-meson 2-particle

continuum cMM , and the parity even bound states c2 and c4. The cMM contribution is

(see e.g. [18])

cMM(s) = 12π

∫ θ

0

dθ

4π

1

4 cosh4 θ2

|FΘb1b1

(θ)|2 , θ ≡ 2 arccosh

√s

2(6.1)

where FΘb1b1

is the form factor of the trace of the stress tensor with the pion-pion two

particle continuum

FΘb1b1

(θ) =2 cos πν

2

cosπν − cosh θcosh

(iπ − θ

2

)exp

(∫2

t

cosh(ν − 1

2

)sinh t cosh t

2

sin2 (iπ − θ)t2π

).

(6.2)

The total contribution is cMM(∞) = 0.235420. The contribution from the bound states

are predicted using the results in [19, 20]

m(SG)2

mgap

= 1.90211,m

(LCT)2

mgap

= 1.90421, c(SG)2 = 0.724596, c

(LCT)2 = 0.729811,

m(SG)4

mgap

= 3.07768,m

(LCT)4

mgap

= 3.09220, c(SG)4 = 0.016748, c

(LCT)4 = 0.016592 .

(6.3)

See Appendix C for details. The combination of cMM , c2 and c4 is accurate up to the

soliton-anti-soliton threshold s =(mgap

sin πν2

)2

= 10.47×m2gap. For energy greater than the

threshold, the above result is still a good approximation since they add up to 96% of the

central charge. On the truncation side, we use (3.11) to compute the spectral density

of T−−. The poles at s = m22 and m2

4 are identified as the eigenstates nearest to the

predicted mi with non-zero contribution to the c-function. The rest of the states belong

to the continuum. When mq is comparable with the UV scale g√Nc/π, the IR physics

gets a significant correction. The behavior of the spectral density is different from that

of the Sine-Gordon model. Some of the Sine-Gordon bound states are no longer stable

due to the interaction with UV degrees of freedom, and they form resonances in the

multi-particle continuum.

Because of the truncation of the Hilbert space, the spectrum of eigenvalues is

discrete and a literal implementation of the spectral density (3.11) produces a discrete

approximation of the true c-function. We can interpolate the c-function by taking

advantage of the analyticity of the time-ordered correlator [21]. First, we use the

– 20 –

discrete spectral density ρT−−(µi) to compute the time-ordered correlator

G(s) =∑i

ρT−−(µi)

s− µ2i + iε

. (6.4)

Away from physical poles and branch cuts, the time-ordered correlator is an analytic

function of s, and its Taylor coefficients in s around, for instance, s = 0, converge as

∆max →∞. From the Taylor series and the analytic properties of the correlator, we can

reconstruct its behavior in the full complex plane. In practice, we take the coordinate

transformation

s = 4m2gap

4z

(1 + z)2(6.5)

so the multi-particle branch cut is mapped to the unit circle. Moreover, G(s) is a

sum of isolated physical poles, with truncation eigenvalues µi < 2mgap, plus a function

of z that is analytic in the unit disk. The basic idea is to approximate G(s) (minus

the stable particle poles) by a Pade approximant in the variable z, denoted as G(s).

Then, we map back to s-coordinate and take the imaginary part Im G(s) to obtain the

interpolation of our spectral density

ρT−−(s) = − 1

πIm G(s) . (6.6)

We show the comparison between the LCT result and the Sine-Gordon prediction for

the c-function in Figure 10.

At larger values of mq, there are unstable particles that appear as resonances in the

spectral density. In this case, it is important to improve the performance of the Pade

approximant by first subtracting out bound states and resonances from G(s), and then

taking a Pade approximant of the remainder.6 Subtracting out bound states is trivial

since they appear directly in the eigenspectrum of the Hamiltonian and consequently are

easily isolated individual terms in the sum over states in (3.11). Removing resonances

is a more involved process. To identify the contribution from the resonance, we first

take the raw spectral density (3.11) from truncation and look at the integrated spectral

density I(s) ≡∫ s

0ds′ρT−−(s′); because of discreteness, this is a sum of step functions.

6The function G(s) minus the stable particle poles is still an analytic function of z in the unitdisk even when there are complex poles from unstable resonances. The decay width of the resonancespushes their poles down in the complex plane through the multiparticle branch cut onto the secondsheet in s, which is outside the unit disk in z. So, in principle with high enough ∆max, the Padeapproximation would automatically capture the resonance bump in the physical regime. However, atfinite ∆max, the Pade approximation procedure works much more efficiently if we fit the Breit-Wignerresonance and treat it separately.

– 21 –

The resonance is a prominent feature where I(s) rises suddenly and steeply. We fit

this behavior to an integrated Breit-Wigner form:

IBW(s) ≡ c0 + a

1

2−

tan−1(M2−sMΓ

)π

, (6.7)

where we obtain the parameters a, c0,M and Γ from the fit. Finally, we want to perform

a Pade approximation to the time-ordered correlator after removing the contribution

from the resonance:

GBW(s) ≡∫ ∞

4m2gap

dµ2 ρBW(µ2)

s− µ2 + iε, ρBW(µ2) ≡ a

MΓ

(s−M2)2 +M2Γ2. (6.8)

To summarize, the final object that we Pade approximate is the time-ordered correlator

with bound states and the resonance removed:

Gsmooth(s) ≡

( ∑i 6=bound

ρT−−(µi)

s− µ2 + iε

)−GBW(s). (6.9)

In practice, we have found that a (4, 4) Pade approximant in the variable z works well.

At that point, we have the spectral density decomposed into three different pieces: the

bound states, which we know directly from the Hamiltonian eigenstates; the resonance,

which we have in the form ρBW(µ2) with the parameters a,M, and Γ from the fit of

(6.7); and the remainder, which we know from ρT−−;smooth ≡ − 1π

Im Gsmooth(s). Adding

these three contributions back together gives us the full spectral density. We show the

result for the spectral density ρT−− at mq = 0.125 in Fig. 11.

7 Conclusions and Future Directions

Gauge theories in two dimensions are remarkably rich and capture many important

features of their higher dimensional cousins. The case we have focused on in this work

was motivated in particular by its similarity to real-world QCD – with Nc = 3, the

theory is nonintegrable, and with small but nonvanishing quark mass mq, there is a

“pion” that is parametrically light compared to the confinement scale. Moreover, the

pion is well-described by a chiral Lagrangian, which is also nonintegrable due to the

presence of an infinite series of effective interactions at all orders in mq, but approaches

the Sine-Gordon model at very small mq. Although still a caricature of our world, it is

nevertheless a recognizable one [22].

– 22 –

� � � ������

�����

�����

�

��

� /�π

ρ�--

� � � � � ����������������������

� /�π

�(�)

Figure 11: Spectral density for T−− from truncation, at mq = 0.125 and Nc = 3. Thebound state δ function has been given a very small width to make it visible to theeye. The multi-particle continuum and resonance peak require processing the discretespectral results from truncation as described in the text. In the inset, we show thecomparison of the integrated spectral density, which in this case is the C-function, fromthe raw LCT data (black dots) and from integrating the processed spectral density (red,solid).

The major technical challenge solved in this work is that a small nonzero quark

mass causes the eigenfunctions of the Hamiltonian to behave nonanalytically, ∼ xα

with 0 < α < 1, at small parton-x. To obtain accurate numeric results, we needed a

basis that accommodated this boundary behavior, in order to achieve fast convergence

as the number of basis states increases, and moreover a method for efficiently comput-

ing Hamiltonian matrix elements in this basis. The method developed here used the

fact that in the free fermion UV CFT, one can treat the nonlocal operator ∂αψ as a

generalized free field, and use it to construct composite primary operators and their

corresponding basis states in Lightcone Conformal Truncation (LCT). The resulting

basis was remarkably efficient, not only in terms of the mass eigenstates but also in

terms of how accurately they were able to capture the correct correlators in the IR.

In particular, as shown in Fig. 10, our Pade approximation of the correlators was ex-

tremely accurate, significantly better than the raw truncation spectral density would

– 23 –

suggest is possible.

Despite the focus here on one particular model, namely that of an SU(3) gauge

group with a single Dirac fermion in the fundamental representation, perhaps the most

interesting application of these methods is to the vast range of IR CFTs that can be

obtained with vanishing quark masses by varying the gauge group and matter content.

The IR has been conjectured [5, 9–11] to be described by gauged WZW coset models,

where the specific coset is a quotient of the global symmetries of the UV free fermion

theory divided by the gauge group. Gauged WZW coset models are extremely versatile

and can be used to construct many well-known CFTs, including minimal models and

much more. By turning on small quark masses, one can also study relevant deformations

of such models, as we did in this work for the compact free boson.

In fact, if one is just interested in the dynamics of the light sector when mq is

small, it seems very likely that one could simplify and improve the method used here

by taking mq → 0 and g →∞ while keeping the physical light mass mπ ∼√mqg fixed.

The results in this work with very small numeric values of mq indicate that this is a

well-behaved limit, and so it would be interesting and useful to work out the rules for

basis states and their Hamiltonian matrix elements in this limit directly.

Acknowledgments

We thank Zuhair Khandker, Silviu Pufu, Matthew Walters, and Xi Yin for helpful

conversations, and Matthew Walters for collaboration at an early stage of the project;

we also thank Igor Klebanov and Matthew Walters for comments on an earlier draft.

ALF and EK were supported in part by the US Department of Energy Office of Science

under Award Number DE-SC0015845, NA, ALF and EK were supported in part by the

Simons Collaboration Grant on the Non-Perturbative Bootstrap, and ALF in part by a

Sloan Foundation fellowship. YX was supported by a Yale Mossman Prize Fellowship

in Physics.

A Heff Derivation of Induced Gauge Interaction

The correct prescription for integrating out the gauge field in lightcone gauge is some-

what subtle. In ’t Hooft’s original paper [12] on his solution to the infinite Nc limit

of the theory, he derived the ‘principal value’ prescription (3.6) by solving the Bethe-

Salpeter equations that emerged from an analysis of Feynman diagrams, using a hard

IR cutoff on momentum |k| > |kmin|. He found that all physical quantities remained

finite in the limit kmin → 0, and his principal value prescription emerged naturally.

Nevertheless, one might wonder whether this prescription is still the correct one at

– 24 –

finite Nc, and in fact some of the literature that followed debated the issue. In any

case, in this appendix we will take the opportunity to show how ’t Hooft’s principal

value prescription emerges from the procedure in [16] for an effective lightcone Hamil-

tonian Heff that results from integrating out zero modes that are discarded in lightcone

quantization.

The basic idea of the effective Hamiltonian Heff is that it follows from match-

ing the time-dependence of correlators computed in equal-time and lightcone quan-

tization, since the correlators are physical and should agree in either quantization

scheme. In particular, given the correlators, one can extract the Hamiltonian ma-

trix elements in either quantization scheme by using the relation 〈O, ti|H|O′, ti〉 =

limtf→ti i∂tf 〈O, ti|O′, tf〉. By using the Dyson series for the unitary time evolution op-

erator U(t, 0) = 1− i∫ t

0dt1H(t1)− 1

2

∫ t0dt1dt2T {H(t1)H(t2)}+ . . . , one can explicitly

see that δ(x+) terms in correlators can produce terms that show up in the lightcone

Hamiltonian but not the equal-time Hamiltonian (because eg they collapse the double

integral in the second order term in the Dyson series to a single integral, but only in

lightcone quantization). See [16], or appendix B of [7], for more details.

With this prescription in hand, the key point is to look at the gauge boson propa-

gator for A+ in equal-time quantization in lightcone gauge:

GET(x) =1

2π

x−

−x+ + iεsgn(x−)=

1

2π

(−P.V.x

−

x+− iπδ(x+)|x−|

). (A.1)

When we integrate out A+, the term that must be added to the effective lightcone

Hamiltonian Heff in order to match correlators computed in equal-time quantization is

the coefficient of the δ(x+) term in the gauge boson propagator. That is, we should

interpret 1∂2

in position space as f(x) 1∂2g(x) = − i

2

∫dy−f(x)|x− − y−|g(y). It is then

straightforward to Fourier transform |x−| and see that this rule corresponds in momen-

tum space to ’t Hooft’s principal value prescription (3.6).

B Details of Modified Basis

We will start from the building blocks ∂αψ, with dimension α+ 12. When using double-

trace construction

A↔∂`B ≡

∑k=0

C`k(∆A,∆B)∂`−kA∂kB , (B.1)

– 25 –

we modify the dimensions ∆A and ∆B that go into the coefficients

C`k(∆A,∆B) ≡ (−1)kΓ (`+ 2∆A) Γ (`+ 2∆B)

k!(`− k)!Γ (k + 2∆B) Γ (−k + `+ 2∆A), (B.2)

where

∆A,∆B = (degree) + n×(α +

1

2

). (B.3)

For example, a neutral two fermion operator in QCD with degree ` can be expressed

as

[∂αψ†∂αψ]` ≡ ∂αψ†P(2α,2α)`

(−→∂ −

←−∂)∂αψ

= µ(2α,2α)∑k

C`k(α + 1/2, α + 1/2)∂α+`−kψ†∂α+kψ . (B.4)

In both radial quantization and Wick contraction (B.4) should give orthogonal states.

This is the basis of the Generalized Free Field theory.

50 100 200

10-9

10-6

0.001

ℓin

δ⟨·|ℳ

|·⟩

α=1/2, ℓ=ℓ'=5

α=1/2, ℓ=ℓ'=20

α=1/2, ℓ=0, ℓ'=20

α=1/1000, ℓ=ℓ'=20

Figure 12: The consecutive difference between individual matrix elements at `in and(`in + 2), as a function of `in. The matrix elements converge as power law. Small αtends to have better accuracy. The accuracy of low ` is better than higher `.

In computing the interaction term matrix elements with respect to the large Nc

GFF basis, we encounter the integral

〈P (2α,2α)` |M|P (2α,2α)

`′ 〉 ≡ P∫dxdx′

[xα(1− x)αx′α(1− x′)αP (2α,2α)

` (1− 2x)P(2α,2α)`′ (1− 2x′)

− xα(1− x)αP(2α,2α)` (1− 2x)P

(2α,2α)`′ (1− 2x)

]1

(x− x′)2,

(B.5)

– 26 –

and we do not have its closed-form expression. We can approximately compute them

by separating the wave function into pieces

xα(1− x)αP(2α,2α)` (1− 2x) = P

(2α,2α)` (1)xα + P

(2α,2α)` (−1)(1− x)α

+∞∑k

ck` P(0,0)k (1− 2x) (B.6)

where the first two pieces are monomials that carry the boundary condition. The re-

maining piece has trivial boundary condition, which we expand as Legendre polynomials

with coefficients

ck` ≡∫ 1

0

dx xα(1− x)αP(2α,2α)` (1− 2x)P

(0,0)k (1− 2x)

− P (2α,2α)` (1)xαP

(0,0)k (1− 2x)

− P (2α,2α)` (−1) (1− x)αP

(0,0)k (1− 2x) , (B.7)

where the boundary-to-Lagendre term has closed-form expression∫ 1

0

dx xαP(0,0)k (1− 2x) =

√2k + 1 (−α)k(α + 1)k+1

(B.8)

and the Jacobi-to-Legendre term is computed as a sum over pairs of monomials∫ 1

0

dx xm1+m2xα(1− x)α =Γ (α + 1) Γ (α +m1 +m2 + 1)

Γ (2α +m1 +m2 + 2)(B.9)

weighted by the coefficients in the Jacobi and Legendre polynomials.

Now we split the matrix element (B.5) in terms of the pieces in (B.6). We gather

like terms using the symmetry of Jacobi polynomials, and the final result is

〈P (2α,2α)` |M|P (2α,2α)

`′ 〉 =(P

(2α,2α)` (1)P

(2α,2α)`′ (1) + P

(2α,2α)` (−1)P

(2α,2α)`′ (−1)

)〈xα|M|xα〉

+(P

(2α,2α)` (1)P

(2α,2α)`′ (−1) + P

(2α,2α)` (−1)P

(2α,2α)`′ (1)

)〈xα|M|(1− x)α〉

+∑k′

ck′

`′

(P

(2α,2α)` (1) + (−1)k

′P

(2α,2α)` (−1)

)〈xα|M|P (0,0)

k′ 〉

+∑k

ck`

(P

(2α,2α)`′ (1) + (−1)kP

(2α,2α)`′ (−1)

)〈xα|M|P (0,0)

k 〉

+∑k

∑k′

ck` ck′

`′ 〈P(0,0)k |M|P (0,0)

k′ 〉 . (B.10)

– 27 –

All integrals that show up in the above formula have closed-form expression

Mk,k′ ≡〈P (0,0)k |M|P (0,0)

k′ 〉

=P∫dxdx′

P(0,0)` (1− 2x)P

(0,0)`′ (1− 2x′)− P (0,0)

` (1− 2x)P(0,0)`′ (1− 2x)

(x− x′)2

= −H(k+k′)/2 −H(−km+kM−1)/2 +H(kM−1)/2 +HkM/2 (B.11)

I1(a, b) ≡〈xa|M|xb〉 = P∫dxdx′

xax′b − xa+b

(x− x′)2

=aHa + bHb − 1

a+ b−Ha+b−1 (B.12)

I1(0, 0) = 0 (B.13)

I2(a, b) ≡〈xa|M|(1− x)b〉 = P∫dxdx′

xa(1− x′)b − xa(1− x)b

(x− x′)2

=1

2a(a+ 1)Γ(a+ b+ 1)

×(− 2a(a+ 1)Γ(b+ 1)Γ(a+ b+ 2)− 2Γ(a+ 2)Γ(b)(abψ(0)(b) + a+ b)

+ 2Γ(a+ b+ 1)∂

∂t

(3F2(2,−a, b+ 1; b+ 2, t; 1)

)∣∣∣∣t=2

)(B.14)

I2(0, b) = I2(a, 0) = 0 (B.15)

where ψ(0)(b) is the digamma function, 3F2 is the regularized hypergeometric function,

and Hn is the n-the harmonic number.

In practice we need to truncate the infinite sums over k and k′ to some “internal

cutoff” `in. We study the convergence of individual matrix elements in Figure 12. We

see that the matrix elements all converge quickly to the exact value. At ∆max = 21+2α,

we can compute the matrix elements to `in = 400 in a few minutes, and the top matrix

element have about 4 digits of accuracy. The fact that low ` matrix elements converge

faster than high `, and that higher ` basis states decouple from the low-lying spectrum

indicate that the precision of the eigenvalues may be better. We conclude that at

∆max = 21+2α and `in = 400 the Hamiltonian is reliable up to a small numerical error.

– 28 –

The fermion mass term has closed-form expression

〈P (2α,2α)` |M(mass)|P (2α,2α)

`′ 〉 =

∫dx xα−1(1− x)α−1P

(2α,2α)` (1− 2x)P

(2α,2α)`′ (1− 2x)

=

{2Γ(2α+`m+1)Γ(2α+`M+1)

2αΓ(`m+1)Γ(4α+`M+1)` = `′ mod 2

0 otherwise, (B.16)

where `M ≡ max(`, `′) and `m ≡ min(`, `′).

The above algorithm computes the matrix elements at large-Nc, and can be easily

generalized to the finite-Nc case. At finite-Nc, we encounter matrix elements with

generic particle numbers. Following the procedure of [7], we arrive at an expression

Mkk′

2p(2π)δ(p− p′)=∑

a,b,a′,b′

2π2p∆+∆′−1A(a,b,a′,b′)

k,k′

Γ(∆ + ∆′ − 1)I(a, b, a′, b′) , (B.17)

where A(a,b,a′,b′)

k,k′comes from spectators and I(a, b, a′, b′) comes from the active part.

With the ∂αψ basis, the powers a, b, a′, b′ are no longer integers but integers shifted

uniformly by α. For the spectator factor A(a,b,a′,b′)

k,k′and many cases in I(a, b, a′, b′),

the expression only contains Gamma functions, and the continuation to non-integers is

straightforward. For most the generic I(a, b, a′, b′), we need to compute

I(a, b, a′, b′) =Γ(∆ + ∆′ − 1)

Γ(a)Γ(b)Γ(a′)Γ(b′)Γ(c)

∫dx1x

a+b+a′+b′−41 (1− x1)c−1

×∫dx2dx3

xa−12 (1− x2)a

′−1(xb−1

3 (1− x3)b′−1 − xb−1

2 (1− x2)b′−1)

(x2 − x3)2.

(B.18)

Directly evaluating (B.18) by brute force for non-integer powers of x2, x3 can be compu-

tationally expensive. However, notice that the Jacobi polynomials xα(1−x)αP(2s,2s)` (1−

2x) form a complete basis of polynomials on x ∈ [0, 1] with boundary condition

xα(1− x)α. So all possible x2, x3 integrals in (B.18) can be written as a linear combi-

nation of (B.5), and there is actually nothing new to compute.

– 29 –

C Form factors in Sine-Gordon theory

In this appendix we show the details in the bound state form factors in (6.3). We

comput the c-function from the spectral density of the stress tensor

c(µ) = 12π

∫ µ2

0

dsρΘ(s)

s2. (C.1)

We focus on the contribution from the k-th bound state bk. The spectral density of

one-particle states is ρk = |FΘbk|2δ(s−mk), and mk = 2ms

(sin π

2kν)4

, so we have

ck =12π|FΘ

bk|2(

2 sin π2kν)4 (C.2)

where we set the soliton mass ms = 1. The form factors of bound states bk can be

extracted from the pole of the 〈ss|Θ|0〉 form factor, divided by the residue of the pole

of the bound state in the b1b2 → b1b2 S-matrix [19]:

FΘbk

= Resθ→θk

FΘss ×√

2

(2iResθ→θk

S+

)− 12

, (C.3)

where the bound state bk is associated with a pole at θk = iπ(1− kν). The parity-even

ss-S-matrix has an integral expression

S+ =sinh iπ

ν+ sinh iπx

ν

sinh iπ(1−x)ν

∫dt

t

sinh(

12(1− ν)t

)sinh(tx)

cosh t2

sinh νt2

. (C.4)

The formulas for FΘss can be found in [20]:

FΘss(θ) =

2√

2i

νsinh

(θ

2

)sinh 1

2(iπ − θ)

sinh 12ν

(iπ − θ)fminss (θ) , (C.5)

where

fminss (θ) =

dt

t

sinh(

12(1− ν)t

)(1− cosh t(1− x))

2 sinh t cosh t2

sinh νt2

. (C.6)

One can check that fminss is regular at θ → θk, so the residue is

ResFΘss(θ) = −2

√2ik+1 sin(πkν)fmin

ss (θ) . (C.7)

– 30 –

The stress tensor has support at even k. At Nc = 3, ν = 15, and we have c2 and c4

contribution to the c-function.

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– 32 –